arXiv:1307.1257v1 [math.AP] 4 Jul 2013
SURFACE PARALLEL TO THE BOUNDARY: STABILITY OF THE RADIAL CONFIGURATION
GIULIO CIRAOLO, ROLANDO MAGNANINI, AND SHIGERU SAKAGUCHI Abstract. Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equa-tions must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the do-main. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is al-most radially symmetric, in the sense that is contained in and contains two concentric balls Bre and Bri, with the difference re−ri (linearly)
controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed by Aftalion, Busca and Reichel in [ABR].
1. Introduction
Let Ω be a bounded domain in RN. It has been noticed in [MS2]-[MS3], [Sh] and [CMS] that positive solutions of homogeneous Dirichlet boundary value problems or initial-boundary value problems for certain elliptic or, respectively, parabolic equations must be radially symmetric (and the un-derlying domain Ω be a ball) if just one of their level surfaces is parallel to ∂Ω (that is, if the distance of its points from ∂Ω remains constant).
The property was first identified in [MS2], motivated by the study of invariant isothermic surfaces of a nonlinear non-degenerate fast diffusion equation (designed upon the heat equation), and was used to extend to nonlinear equations the symmetry results obtained in [MS1] for the heat equation. The proof hinges on the method of moving planes developed by J. Serrin in [Se] upon A.D. Aleksandrov’s reflection principle ([Al]). Under slightly different assumptions and by a different proof still based on the method of moving planes, a similar result was independently obtained in [Sh] (see also [GGS]). Further extensions can be found in [MS3] and [CMS]. To clarify matters, we consider a simple situation (possibly the simplest). Let G be a C1-smooth domain in RN, denote by B
Rthe ball centered at the
origin with radius R, 1 and define the set
Ω = G + BR= {y + z : y ∈ D, |z| < R}
1991 Mathematics Subject Classification. Primary 35B06, 35J05, 35J61; Secondary 35B35, 35B09.
Key words and phrases. Parallel surfaces, overdetermined problems, method of moving planes, stability, stationary surfaces, Harnack’s inequality.
1In the sequel, a ball with radius R centered at a generic point x will be denoted by
BR(x).
— the Minkowski sum of G and BR. Consider the solution u = u(x) of the
torsion boundary value problem
− ∆u = 1 in Ω, u = 0 on ∂Ω. (1.1)
As shown in the aforementioned references, if there is a positive constant c such that
u = c on ∂G, (1.2)
then G and Ω must be concentric balls.
The aim of this paper is to investigate on the stability of the radial con-figuration. In other words, we suppose that u is almost constant on ∂G by assuming that the semi-norm
[u]∂G= sup x,y∈∂G,
x6=y
|u(x) − u(y)| |x − y|
is small and want to quantitatively show that Ω is close to a ball. A result in this direction that concerns the torsion problem (1.1) is the following theorem.
Theorem 1.1. Let G be a bounded domain in RN with boundary ∂G of class C2,α, 0 < α ≤ 1, and set Ω = G+B
Rfor some R > 0. Let u ∈ C2(Ω)∩C0(Ω)
be the solution of (1.1).
There exist constants ε, C > 0 such that, if [u]∂G ≤ ε,
then there are two concentric balls Bri and Bre such that
Bri ⊂ Ω ⊂ Bre and
(1.3)
re− ri≤ C [u]∂G.
(1.4)
The constants ε and C only depend on N , the C2,α-regularity of ∂G, the diameter of G and, more importantly, the number R.
A result similar to Theorem 1.1 was proved by Aftalion, Busca and Re-ichel in [ABR]. There, under further suitable assumptions, it is proved the stability estimate (1.5) re− ri ≤ C | log uν − dkC1(∂Ω) −1/N,2
where uν denotes the (exterior) normal derivative of the solution of (1.1),
d is a constant and k · kC1(∂Ω) is the usual C1 norm on ∂G; (1.5) is the
quantitative version of Serrin’s symmetry result [Se] that stated that the solutions of (1.1) whose normal derivative is constant on ∂Ω, that is
(1.6) uν = d on ∂Ω,
must be radially symmetric (and Ω must be a ball). In [ABR], (1.5) is a par-ticular case of a similar estimate that holds for general semilinear equations, that is when solutions of the problem
(1.7) − ∆u = f (u) and u ≥ 0 in Ω, u = 0 on ∂Ω, are considered (here, f is a locally Lipschitz continuous function).
The logarithmic dependence in (1.5) is due to the method of proof em-ployed, which is based on the idea of refining the method of moving planes from a quantitative point of view. As that method is based on the maximum (or comparison) principle, its quantitative counterpart is based on Harnack’s inequality and some quantitative versions of Hopf ’s boundary lemma and Serrin’s corner lemma (that involves the second derivatives of the solution on ∂Ω). The exponential dependence of the constant involved in Harnack’s inequality leads to the logarithmic dependence in (1.5).
Estimate (1.5) was largely improved in [BNST2] (see also [BNST1]) but only for the case of the torsion problem (1.1) (a version of this result is also available for Monge-Amp`ere equations [BNST3]). There, (1.5) is enhanced in two respects: the logarithmic dependence at the right-hand side of (1.5) is replaced by a power law of H¨older type; an estimate is also given in which the C1-norm is merely replaced by the L1-norm. The results in [BNST1] are ob-tained by remodeling in a quantitative manner the Weinberger’s proof [We] of Serrin’s result, which is grounded on integral formulas such as Rellich-Pohozaev’s identity. We observe that, unfortunately, that approach does not seem to work in our setting, since the overdetermining condition (1.2) does not conveniently match the underlying integral formulas of variational type.
The proof of Theorem 1.1 adapts the approach used in [ABR] but, dif-ferently from that paper, it results in the linear estimate (1.3)-(1.4). The reason for this gain must be ascribed to the fact that, in our setting, we deal with the values of u on ∂G, which is in the interior of Ω, and we are dispensed with the use of estimates up to the boundary of ∂Ω, which are accountable for the logarithmic behavior in (1.5). However, the bene-fit obtained in Theorem 1.1 has a cost: the constant C in (1.4) blows up exponentially to infinity as R tends to zero.
The proof can be outlined as follows. For any fixed direction ω, the method of moving planes determines a hyperplane π, orthogonal to ω and in critical3position, and a domain X contained in G and symmetric about π. Since X ⊆ G ⊂ Ω, we can use interior and boundary Harnack’s inequalities to estimate in terms of [u]∂G the values of the harmonic function w(x) =
u(x′)−u(x) (here, x′is the mirror image in π of the point x) in the half of X
where w is non-negative. Such an estimate gives a bound on the distance of ∂X from ∂G; it is important to observe that such a bound does not depend on the direction ω. By repeating this argument for N orthogonal directions, an approximate center of symmetry — where the two balls Bri and Bre in
(1.3) are centered — can be determined. The radii of these balls then satisfy (1.4).
We conclude this introduction with some remarks on the relationship between problem (1.1)-(1.2) and Serrin’s problem (1.1)-(1.6). In both cases, a solution exists (if and) only if the domain is a ball; a natural question arises: does the symmetry obtained for one of these problems imply that for the other one?
We begin by observing that conditions (1.1)-(1.2) neither imply nor are implied by conditions (1.1)-(1.6), directly. However, we can claim that (1.6)
seems to be a stronger constraint, in the sense that, in order to obtain it, one has to require that (1.2) holds (with different constants) for at least a sequence of parallel surfaces clustering around ∂Ω. On the other hand, if (1.1)-(1.6) holds, we cannot claim that its solution u is constant on surfaces parallel to ∂Ω, but we can surely say that the oscillation of u on a parallel surface becomes smaller, the closer the surface is to ∂Ω; in fact, an easy Taylor-expansion argument tells us that, if Γt= {x ∈ Ω : dist(x, ∂Ω) = t},
then (1.8) max Γt u − min Γt u = O(t2) as t → 0+.
Theorem 1.1 suggests the possibility that Serrin’s symmetry result may be obtained, so to speak by stability, via (1.4) (conveniently remodeled) and (1.8) in the limit as t → 0+. If successful, this plan would show that the method of moving planes can be employed to prove Serrin’s symmetry re-sult with no need of the aforementioned corner’s lemma and under weaker assumptions on ∂Ω and u (see also [Sh]). To our knowledge, this issue has been addressed only in [Pr], for domains with one possible corner or cusp, and in [GL], by arguments based on [We]. At this moment, the blowing up dependence of the constant C in (1.4) on the distance of the relevant parallel surface from ∂Ω prevents us to make this possibility come true.
The paper is organized as follows. In Section 2 we introduce some no-tation and, for the reader’s convenience, we recall the proof of the relevant symmetry result. In Section 3, we carry out the proof of Theorem 1.1. In Section 4, we explain how our result can be extended to the classical case of semilinear equations (1.7).
2. Parallel surfaces and symmetry
To assist the reader to understand the proof of Theorem 1.1, here we summarize the arguments developed in [MS2], [MS3] and [CMS] to prove the symmetry of a domain Ω admitting a solution u of (1.1) and (1.2). This will be the occasion to set up some necessary notations.
In this section we assume that the boundary of G is of class C1; this assumption guarantees that the method of moving planes can be initialized (see [Fr]).
Let ω ∈ RN be a unit vector and λ ∈ R a parameter. We define the
following objects:
(2.1)
πλ = {x ∈ RN : x · ω = λ} a hyperplane orthogonal to ω,
Aλ = {x ∈ A : x · ω > λ} the right-hand cap of a set A,
xλ = x − 2(x · ω − λ) ω the reflection of x about πλ,
Aλ = {x ∈ RN : xλ ∈ Aλ} the reflected cap about πλ.
Set M = sup{x · ω : x ∈ G}, the extent of G in direction ω; if λ < M is close to M , the reflected cap Gλ is contained in G (see [Fr]). Set
(2.2) m = inf{µ : Gλ ⊂ G for all λ ∈ (µ, M )}. Then for λ = m at least one of the following two cases occurs:
(i) Gλ becomes internally tangent to ∂G at some point P ∈ ∂G \ πλ;
We notice that if
Ω = G + BR= {x + y : x ∈ G, y ∈ BR},
then we have that
Gλ ⊂ G ⇒ Ωλ ⊂ Ω,
which follows from an easy adaptation of [MS3, Lemmas 2.1 and 2.2]. Theorem 2.1. Let G be a bounded domain in RN with boundary ∂G of class
C1 and set Ω = G + BR for some R > 0. Suppose there exists a function
u ∈ C0(Ω) ∩ C2(Ω) satisfying (1.1) and (1.2).
Then G (and hence Ω) is a ball and u is radially symmetric.4
Proof. By contradiction, let us assume that G is not a ball. Let ω ∈ SN −1 and, on the right-hand cap Ωm at critical position λ = m, consider the
function defined by
(2.3) wm(x) = u(xm) − u(x), x ∈ Ωm.
It is clear that wm satisfies the conditions
(2.4) ∆wm= 0 and wm≥ 0 on ∂Ωm.
Moreover, since G is not a ball, there is at least one direction ω such that wmis not identically zero on ∂Ωm; hence, by the strong maximum principle,
we have that
(2.5) wm> 0 in Ωm.
Also, we can apply Hopf’s boundary point lemma to the points in ∂Ωm∩
πm and obtain that
∂wm ∂ω > 0 on ∂Ωm∩ πm, and hence (2.6) ∂u ∂ω < 0 on ∂Ωm∩ πm, since ∂wm ∂ω = −2 ∂u ∂ω on Ω ∩ πm.
If (i) holds, we get a contradiction to (2.5) by observing that wm(P ) = u(Pm) − u(P ) = 0,
since (1.2) holds and both P and Pm belong to ∂G.
If (ii) holds, we observe that ω is tangent to ∂G at Q and that (1.2) implies that
∂u
∂ω(Q) = 0, contradicting (2.6).
Thus, we conclude that G must be a ball and hence Ω = G + BR is also
a ball.
4Note that, if G is a spherical annulus in RN, then the solution of (1.1) is radially
3. Stability estimates for the torsion problem
In this section, we present a proof of Theorem 1.1 that relies on the ideas in [ABR] and the Harnack-type and tangential stability estimates contained in Lemmas 3.1 and 3.2, respectively.
These Lemmas analyse the two critical situations (i) and (ii) of Section 2 from a quantitative point of view. Thus, we shall consider two domains in RN, D1 and D2, containing the origin and such that D1 ⊂ D2. We shall
also suppose that there is an R > 0 such that BR(x) ⊂ D2 for any x ∈ D1
and use the following notations:
(3.1) D
+
j = {x ∈ Dj : x1> 0}, j = 1, 2,
ER= {x ∈ D1+ BR/2: x1> R/2}.
Without loss of generality, we can assume that D1+ is connected; this easily implies that also ER is connected.
Lemma 3.1. Assume that a function w ∈ C0(D2) ∩ C2(D2+) satisfies the
conditions
∆w = 0 and w ≥ 0 in D+2, w = 0 on ∂D+2 ∩ {x1 = 0},
and let z = (z1, . . . , zN) be any point in D1+.
Then, for every x ∈ ER it holds that
w(x) ≤ Cw(z) z1 , if z1> 0, (3.2) w(x) ≤ C ∂w ∂x1 (z), if z1 = 0, (3.3)
where C is a constant depending on N , the diameters of D1 and D2, and R
(see (3.10) below).
Proof. We can always assume that w > 0 in D2+. We shall distinguish the three cases: (i) z1 ≥ R/2; (ii) 0 < z1 < R/2; (iii) z1 = 0.
(i) Let Br(x0) be any ball contained in D+2. It is well-known that
Har-nack’s inequality (see [GT]) gives: (3.4) rN −2 r − |x − x0| (r + |x − x0|)N −1 w(x0) ≤ w(x) ≤ rN −2 r + |x − x0| (r − |x − x0|)N −1 w(x0),
for any x ∈ Br(x0). By the monotonicity of the two ratios in |x − x0|, we
have that
(3.5) 2N −231−Nw(x0) ≤ w(x) ≤ 3 · 2N −2w(x0) for any x ∈ Br/2(x0).
Let y ∈ ER be such that
w(y) = sup
ER
w.
Since dist(y, ∂D+2) ≥ R/2 and (D1+ BR/2) ∩ {x1 > s} is connected for any
0 ≤ s ≤ R/2, by [ABR, Lemma A.1], there exists a chain of n pairwise disjoint balls {BR/4(pi)}ni=1 such that dist(pi, ∂D2+) ≥ R/2, i = 1, . . . , n,
y, z ∈
n
[
i=1
and
BR/4(pi) ∩ BR/4(pi+1) 6= ∅, i = 1, . . . , n − 1.
Since BR/2(pi) ⊂ D2+, by applying (3.5) to each ball and combining the
resulting inequalities yields that
(3.6) sup
ER
w ≤ (3 · 2N −2)n+1w(z).
Since z1≤ diam D1+≤ diam D1, we easily obtain that
(3.7) sup ER w ≤ (diam D1)(3 · 2N −2)n+1 w(z) z1 .
An upper bound for the optimal number of balls n is clearly (2 diam D2)NR−N;
the last inequality then implies (3.2).
(ii) Since 0 < z1 < R/2, the point ˆz = (R/2, z2, . . . , zN) is such that
z ∈ BR/2(ˆz) ⊂ D+2.
We apply (3.4) with x0= ˆz, x = z and r = R/2 and obtain that
R 2 N −2 z1 (R − z1)N −1 w(ˆz) ≤ w(z), and hence w(ˆz) ≤ 2N −2Rw(z) z1 .
By noticing that ˆz ∈ ER, we can use the same Harnack-type argument used
for case (i) (see formula (3.6)) to obtain sup ER w ≤ (3 · 2N −2)[1+(2 diam D2)N/RN]w(ˆz), and hence (3.8) sup ER w ≤ 2N −2R(3 · 2N −2)[1+(2 diam D2)N/RN]w(z) z1 .
(iii) By taking the limit as z1 → 0 in (3.8) and noticing that w = 0 for
x1 = 0, we obtain (3.3) as (3.9) sup ER w ≤ 2N −2R(3 · 2N −2)[1+(2 diam D2)N/RN]∂w ∂x1 (z).
An inspection of (3.7), (3.8) and (3.9) informs us that the constant C in (3.2) and (3.3) is given by
(3.10)
C = 3 max(2N −2R, diam D1)2N −2C(diam D2/R) N
N with CN = 32
N
2(N −2)2N. Now, we extend estimates (3.2) and (3.3) to the case in which x is any point in D+1 \ {x1 = 0}.
Lemma 3.2. Let D1, D2, R, w and z be as in Lemma 3.1.
Then, for any x ∈ D1+\ {x1= 0}, it holds that
w(x) ≤ M C w(z) z1 , if z1 > 0, w(x) ≤ M C ∂w ∂x1 (z), if z1 = 0,
where C is given by (3.10) and M is a constant only depending on N . Proof. For x ∈ D1+∩ ERwe clearly have that w(x) ≤ supERw.
If x ∈ D+1 and 0 < x1 < R/2, we estimate w(x) in terms of the value w(ˆx)
with ˆx = (R/2, x2, . . . , xN) ∈ ER. In order to do this, we use the following
boundary Harnack’s inequality (see for instance [CS, Theorem 11.5]):
(3.11) sup
B+R/2(x0)
w ≤ M w(ˆx),
with x0 = (0, x2, . . . , xN); here, M ≥ 1 is a constant only depending N .
Since M ≥ 1 and w(ˆx) ≤ supERw, Lemma 3.1 yields the conclusion. We notice that since ∂G is of class C2,α, then it satisfies a uniform interior ball condition of (optimal) radius ρ > 0 at each point of the boundary. The following result is our analog of [ABR, Proposition 1].
Theorem 3.3. Let Ω and G be as in Theorem 1.1 and consider a solution u ∈ C2,α(Ω) of (1.1).
Pick a unit vector ω ∈ RN and let Gm be the maximal cap of G in the
direction ω as defined by (2.1) and (2.2). Then for (a connected component of ) Gm we have that
(3.12) wm ≤ M C [u]∂G in Gm;
here, wm is defined by (2.3) and C is a constant depending on N , R, the
diameter of G and the C2-regularity of ∂G (see (3.15) below).
Proof. We adopt the same notations introduced for the proof of Theorem 2.1 and we can always assume that ω = e1. The function w = wm satisfies
(2.4) and
w = 0 on ∂Gm∩ πm.
Let us assume that case (i) of the proof of Theorem 2.1 occurs. Without loss of generality, we shall denote by the same symbols Gm and Ωm,
respec-tively, the connected component of Gm which intersects BR(Pm) and the
corresponding connected component of Ωm. Modulo a translation in the
direction of e1, we can apply Lemma 3.2 with D+1 = Gm and D2+= Ωm and
z = Pm. We obtain the estimate
(3.13) w ≤ M C′ w(P
m)
dist(Pm, π m)
in Gm,
where C′ is computed by means of (3.10), that is
C′ = max(2N −2R, diam G) CN(2+diam G/R)N; here, we have used that diam Ω = 2R + diam G.
In order to estimate w(Pm)/dist(Pm, π
m), we notice that |P − Pm| =
2 dist(Pm, πm) and we distinguish two cases.
If |P − Pm| ≥ ρ, since P and Pm lie on ∂G, then
w(Pm) = u(P ) − u(Pm) ≤ diam(G) [u]∂G,
and hence we easily obtain that w(Pm)
dist(Pm, π m)
≤2 diam(G) ρ [u]∂G.
If instead |P − Pm| < ρ, every point of the segment joining P to Pm is at distance less than ρ from a connected component of ∂G. The curve γ obtained by projecting that segment on ∂G has length bounded by C′′|P −
Pm|, where C′′ is a constant depending on ρ and on the regularity of ∂G.
An application of the mean value theorem to the function u restricted to γ gives that u(P ) − u(Pm) can be estimated by the length of γ times the
maximum of the tangential gradient of u on ∂G. Thus, (3.14) w(Pm) ≤ 2 C′′dist(Pm, πm) [u]∂G.
From (3.10) and (3.13)–(3.14) we then get (3.12).
Now, let us assume that case (ii) of the proof of Theorem 2.1 occurs. Again, without loss of generality, we denote by the same symbol Gm the
connected component of Gm which intersects BR(Q), and Ωm accordingly.
Then, we apply Lemma 3.2 with D+1 = Gm, D2+ = Ωm and z = Q, and
obtain that
w(x) ≤ M C ∂w(Q) ∂ω ,
for all x ∈ Gm. Since ω belongs to the tangent hyperplane to ∂G at Q, we
again obtain (3.12).
An inspection of the calculations in this proof informs us that the constant C in (3.12) can be chosen as (3.15) C = max 1,2 diam G ρ , 2ρ C ′′ max R,diam G 2N −2 C[2+ diamG R ]N N . From now on, the proof of our Theorem 1.1 follows the arguments used in [ABR, Sections 3 and 4], with a major change (we use our Theorem 3.3 instead of [ABR, Proposition 1]) and some minor changes that we shall sketch below.
The key idea is to use the smallness of wm in (the relevant connected
component of) Gm to show that G is almost equal to the symmetric open
set X which is defined as the interior of
Gm∪ Gm∪ (∂Gm∩ πm).
In order to do that, we need a priori bounds on u from below in terms of the distance function from ∂G. In [ABR], such bounds are derived in Proposition 4, where it is proven that, if u = 0 on ∂Ω, then there exist positive constants K1 and K2 such that
1/K1 and K2 are bounded by a constant depending on the diameter of Ω
and the C2,α-regularity of ∂Ω.
However, if we examine our situation, we notice two facts: on one hand, differently from [ABR], our symmetrized set X is the reflection of a con-nected component of G — compactly contained in Ω at distance R from ∂Ω — and we need not extend the relevant estimates up to the boundary of Ω, thus obtaining a better rate of stability. On the other hand, since we are dealing with the distance from ∂G instead of that from ∂Ω and u is not constant on ∂G, we need a stability version of the first inequality in (3.16). This is done in the following lemma.
Lemma 3.4. There exists a positive constant K depending on diam G and the C2,α regularity of ∂G such that
(3.17) K dist(x, ∂G) + min ∂G u ≤ u(x), for every x ∈ G. Moreover, if x ∈ ∂X, then (3.18) dist(x, ∂G) ≤ C∗[u]∂G, with (3.19) C∗ = C + diam(G) K ,
and where C is given by (3.15). Proof. Let v be such that
∆v = −1 in G, v = min
∂G u on ∂G;
by the comparison principle, v ≤ u on G. Since v − min∂Gu = 0 on ∂G,
applying (3.16) to ∂G instead of ∂Ω yields that K dist(x, ∂G) ≤ v(x) − min
∂G u for any x ∈ G;
here, we have set K = K1. Since v ≤ u we obtain (3.17).
Now, we prove (3.18). If x · ω ≥ m then x ∈ ∂G and (3.18) clearly holds. If x · ω < m, we notice that xm ∈ ∂G and then, by Theorem 3.3,
u(x) = wm(xm) + u(xm) ≤ C [u]∂G+ u(xm).
Since xm ∈ ∂G, then
u(xm) − min
∂G u ≤ diam(G) [u]∂G;
the last two inequalities then give that u(x) − min
∂G u ≤ (C + diam(G))[u]∂G.
This inequality and (3.17) give (3.18) at once.
For s > 0, let Gsq be the subset of G parallel to ∂G at distance s, i.e.
Gsq = {x ∈ G : dist(x, ∂G) > s}.
We notice that Gs
q is connected for s < ρ/2; indeed, any path in G connecting
any two points x, y ∈ Gsq can be moved inwards into Gsq by the normal field
Next result is crucial to prove Theorem 1.1. Theorem 3.5. Let
[u]∂G <
ρ 4C∗.
Then, for every s in the interval (C∗[u]∂G, ρ/2), we have that
(3.20) Gsq ⊂ X ⊂ G.
Proof. In order to prove (3.20), we proceed by contradiction. Since the maximal cap Gm contains a ball of radius ρ/2, then X intersects Gsq. Let us
assume that there exist a point y ∈ Gsq\ X and take x ∈ X ∩ Gsq. Since Gsq
is connected, we can join x to y with a path contained in Gs
q. Let z be the
first point on this path that falls outside X; we notice that z ∈ ∂X ∩ Gsq.
If z · ω ≥ m, then z ∈ ∂G, that contradicts the fact that dist(z, ∂G) > s. If instead z · ω < m, a contradiction is reached by observing that, by Lemma
3.4, s < dist(z, ∂G) ≤ C∗[u]∂G.
We have now all the ingredients to complete the proof of Theorem 1.1. Proof of Theorem 1.1. An admissible choice of the parameter s in Theorem 3.5 is
(3.21) s = 2C∗[u]∂G.
Theorem 3.5 then implies that, for any x ∈ ∂G, there is a point y ∈ ∂G such that
(3.22) |xm− y| ≤ 2s,
(see [ABR, Corollary 1]). It is important to observe that s does not depend on the direction ω. Thus, we can choose ω to be each one of the coordi-nate directions e1, . . . , eN; by (2.1) and (2.2), these choices define N
hyper-planes πm1, . . . , πmN each in critical position with respect to each direction
e1, . . . , eN, respectively.
Let O be the point of intersection of the N hyperplanes πm1, . . . , πmN and
denote by xO the reflection 2O − x of x in O; since xO is obtained by N successive reflections with respect to the planes πm1, . . . , πmN, by (3.22) one
can infer that for any point x ∈ ∂G there is a point y ∈ ∂G such that |xO− y| ≤ 2N s
(see [ABR, Corollary 2]).
The point O can now be chosen as the center of the balls Bri and Bre in
(1.3). In fact, since (3.22) holds, [ABR, Proposition 6] guarantees that, for any direction ω,
dist(O, πm) ≤ 4N (1 + diam G) s
with s given by (3.21). It is clear that (1.3) holds with ri = min
x∈∂G|x − O| and re= maxx∈∂G|x − O|.
Finally, [ABR, Proposition 7] states that
re− ri ≤ 8N (1 + diam G) s.
Remark 3.6. We notice that the constant C in Theorem 1.1 is given by C = 16N (1 + diam G) C∗.
Hence, the dependence of C on R is of the following type: C = O(AR−N
) as R → 0+, where A > 1 is a constant and
C = O(R) as R → +∞. 4. Semilinear equations
In this section, we shall show how Theorem 1.1 can be extended to the case in which (1.1) is replaced by (1.7). The structure of the proofs is the same: for this reason, we will limit ourselves to identify only the relevant passages that need to be modified. In what follows, we will use the standard norms: kukC1(Ω)= max Ω |u| + max Ω |Du|, kukC2(Ω)= kukC1(Ω)+ max
Ω
|D2u|.
Let f be a locally Lipschitz continuous function with f (0) ≥ 0 and con-sider the following problem
(4.1) ∆u + f (u) = 0, in Ω, u > 0, in Ω, u = 0, on ∂Ω, where Ω = G + BR.
Also in this case, if we add the overdetermination (1.2), then the G must be a ball and u is radially symmetric. The details of this result can be reconstructed from [CMS].
In the following theorem, we assume the domain G as in Theorem 1.1 . Theorem 4.1. Let u ∈ C2(Ω) be the solution of (4.1) with −∂u/∂ν ≥ d0 >
0 on ∂Ω.
If R < 12d0kuk−1C2(Ω), then there exist constants ε, C > 0 such that, if
[u]∂G ≤ ε,
then there are two concentric balls Bri and Bre such that (1.3) and (1.4)
hold, that is
Bri ⊂ Ω ⊂ Bre and
re− ri≤ C [u]∂G.
The constants ε and C only depend on N , R, the C2,α-regularity of ∂G and
on upper bounds for the diameter of G, maxΩ|u| and the Lipschitz constant of f .
As for Theorem 1.1, the proof of Theorem 4.1 is based on a quantitative study of the method of moving planes. In this case, the function wm defined by (2.3) satisfies the conditions
where for x ∈ Ωm c(x) = f (u(xm)) − f (u(x))
u(xm) − u(x) if u(xm) 6= u(x),
0 if u(xm) = u(x).
Notice that c(x) is bounded by the Lipschitz constant L of f in the interval [0, max
Ω u].
The following Lemma summarizes and generalizes the contents of Lemmas 3.1 and 3.2 to the present case.
Lemma 4.2. Let D1, D2 and R be as in Lemma 3.1. Assume that a function
w ∈ C0(D2) ∩ C2(D+2) satisfies the conditions
(
∆w + c(x)w = 0, w ≥ 0, in D2+,
w = 0, on ∂D2+∩ {x1 = 0},
with |c| ≤ L in D2+.
Let z ∈ D+1. Then, for any x ∈ D1+\ {x1 = 0}, it holds that
w(x) ≤ Cw(z) z1 , if z1 > 0, (4.2) w(x) ≤ C∂w ∂x1 (z), if z1 = 0, (4.3)
where C is a constant depending only on N , R, L and the diameters of D1
and D2.
Proof. Let ER be defined by (3.1). First, we prove (4.2) and (4.3) for x ∈
ER (as done in Lemma 3.1) and then we extend such estimates to any
x ∈ D1+\ {x1 = 0} (as in Lemma 3.2). We follow step by step the proofs of
Lemmas 3.1 and 3.2.
The main ingredient in the proof of Lemma 3.1 was Harnack’s inequality. In the present case, [GT, Theorem 8.20] gives
(4.4) sup
Br/4
w ≤ C inf
Br/4
w,
for any Br ⊂ D1+; here, the constant C can be bounded by C √
N +√LR
N , where
CN is a constant depending only on N . Inequality (4.4) replaces (3.5) in
this setting.
If we assume that case (i) of the proof of Lemma 3.1 holds, we readily obtain sup ER w ≤ (diam D1)C(2 diam D2/R) Nw(z) z1 .
If case (ii) or case (iii) hold, the conclusion follows by applying [ABR, Propo-sition 2] and [ABR, Lemma 2], respectively.
To prove the analogous of Lemma 3.2, we need the equivalent of (3.11) for the semilinear case. This can be found, for instance, in [BCN, Theorem 1.3], which gives
sup
B+R/2
here, the constant M depends only on R, N and L. The rest of the proof is
completely analogous to the one of Lemma 3.2.
Theorem 3.3 and its proof apply also to the semilinear case. It is clear that the constant M C appearing in (3.12) will now depend on N , R, diam G, the C2 regularity of ∂G, and L.
Lemma 3.4 is the last ingredient that needs some remodeling.
Lemma 4.3. Let u ∈ C2(Ω) be a solution of (4.1) with −∂u/∂ν ≥ d0 > 0
on ∂Ω, where ν is the normal exterior to ∂Ω.
If R < 12d0kuk−1C2(Ω), then there exists a positive constant K depending
on diam G, maxΩ|u|, the C2,α regularity of ∂G and the Lipschitz constant L such that
K dist(x, ∂G) + min
∂G u ≤ u(x),
for every x ∈ G.
Proof. The proof follows the steps of that of [ABR, Proposition 4]. Let x ∈ G and assume that dist(x, ∂G) < ρ, where ρ is the optimal radius of the interior touching ball to ∂G. For such x, there exist y ∈ ∂G and z ∈ ∂Ω such that
y − x = dist(x, ∂G) ν(y), z − x = (dist(x, ∂G) + R) ν(z).
We notice that ν(y) = ν(z), |y − z| = R and −∇u(z) · ν(z) ≥ d0. The mean
value theorem implies that
(4.5) − ∇u(y) · ν(y) ≥ d0− kukC2(Ω)R ≥
d0
2 ,
for every R ≤ 12d0/kukC2(Ω). Again, by Taylor expansion we have
u(x) ≥ u(y) + ∇u(y) · (x − y) − 1
2kukC2(Ω)|x − y|
2,
and from (4.5) we obtain that u(x) ≥ u(y) +1
4d0|x − y|, for every |x − y| ≤ 1 2d0kuk
−1 C2(Ω),
which implies that
(4.6) u(x) ≥ min
∂G u +
1
4d0dist(x, ∂G), for every x such that dist(x, ∂G) ≤ 12d0kuk−1C2(Ω).
Now, (4.6) replaces formula (28) in the proof of [ABR, Proposition 4] and each argument of that proof can be repeated in our case leading to the
conclusion.
Proof of Theorem 4.1. The proof follows the outline of that of Theorem 1.1. For any fixed direction ω, the moving plane method provides a maximal cap Gm where, thanks to Lemma 4.2, we have the bound
wm ≤ C [u]∂G in Gm,
which is the analog of (3.12). Here, wm is defined by (2.3) and C is a constant depending on N , R, diam G, the C2 regularity of ∂G and on L.
Then, we define a symmetric set X as done in Section 3. We use Lemma 4.3 and prove that
dist(x, ∂G) ≤ C∗[u]∂G,
for any x ∈ ∂X, where C is as in (3.19). This says that, if [u]∂G is small, then X is almost equal to G; in particular we have that
Gsq ⊂ X ⊂ G,
for every s in the interval (C∗[u]∂G, ρ/2) and whenever
[u]∂G <
ρ 4C∗.
We notice that such estimates do not depend on the direction ω. By choos-ing ω to be each of the coordinate directions e1, . . . , eN, we define the
ap-proximate center of symmetry and the conclusion follows by repeating the
argument of the proof of Theorem 1.1.
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Dipartimento di Matematica e Informatica, Universit`a di Palermo, Via Archi-rafi 34, 90123, Italy.
E-mail address: g.ciraolo@math.unipa.it URL: http://www.math.unipa.it/~g.ciraolo/
Dipartimento di Matematica ed Informatica “U. Dini”, Universit`a di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy.
E-mail address: magnanin@math.unifi.it
URL: http://web.math.unifi.it/users/magnanin
Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan.
E-mail address: sigersak@m.tohoku.ac.jp URL: http://researchmap.jp/sigersak2012415/
